The study of optimal transport has a long history dating from the 1781 work of Mongè in [1]} which established what corresponds to the transport map formulation of the subject.
Throughout optimal transport's subsequent development, the problem has been rephrased multiple times, each time revealing a new facet of the theory and spurring new developments and applications.
One particularly important example of this was Kantorovich's reformulation of the problem in 1942 [2]} which cast the problem in terms of joint probability measures by introducing the notion of the optimal coupling.
This enabled optimal transport problems to be formulated as linear programming problems and made them computationally feasible.
